Finite Element Analysis of Cauchy–Born Approximations to Atomistic Models

Abstract

ABSTRACT. This paper is devoted to a new finite element consistency analysis of Cauchy-Born approximations to atomistic models of crystalline materials in two and three space dimensions. Through this approach new "atomistic Cauchy-Born" models are introduced and analyzed. These intermediate models can be seen as first level atomistic/quasicontinuum approximations in the sense that they involve only short-range interactions. The analysis and the models developed herein are expected to be useful in the design of coupled atomistic/continuum methods in more than one dimension. Taking full advantage of the symmetries of the atomistic lattice we show that the consistency error of the models considered both in energies and in dual W 1,p type norms is O(ε 2 ), where ε denotes the interatomic distance in the lattice.

On Atomistic-to-Continuum Couplings without Ghost Forces in Three Dimensions

“…It was shown in [12] that this model is both energy-and variationally consistent to second order in ε, approximating the exact atomistic model as well as the continuum Cauchy-Born model.…”

“…An alternative discrete model defined over cells was introduced in [12]. The average discrete derivatives were defined, e.g., as…”

“…This leads to a discrete gradient ∇y in analogy to (2.3); see [12] for details. The corresponding cell atomistic Cauchy-Born energy is then defined by…”

“…Since the continuum model fails to provide an accurate prediction in the vicinity of defects and other singularities, coupled atomistic/continuum methods have become popular as an adaptive modeling approach over the last years, see e.g. references in [12,14,15]. The main issue that arises in these methods is the proper matching of information across scales.…”

“…In Section 4 we present a conforming coupling method based on bond volumes. We note that in the continuum region we use the atomistic Cauchy-Born models, introduced in [12]. In Section 5 we show that it is possible to introduce discontinuities at the interface, thus allowing greater flexibility in the design of underlying meshes, while still obtaining a consistent ghost-force-free method.…”